When an investor is faced with a portfolio choice problem, the number of possible assets and the various combinations and proportions in which each can be held can seem overwhelming. In this course, you’ll learn the basic principles underlying optimal portfolio construction, diversification, and risk management. You’ll start by acquiring the tools to characterize an investor’s risk and return trade-off. You will next analyze how a portfolio choice problem can be structured and learn how to solve for and implement the optimal portfolio solution. Finally, you will learn about the main pricing models for equilibrium asset prices.
Learners will:
• Develop risk and return measures for portfolio of assets
• Understand the main insights from modern portfolio theory based on diversification
• Describe and identify efficient portfolios that manage risk effectively
• Solve for portfolio with the best risk-return trade-offs
• Understand how risk preference drive optimal asset allocation decisions
• Describe and use equilibrium asset pricing models.

In this module, you will learn about mean-variance optimization: how to make optimal capital allocation and portfolio choice decisions when investors have mean-variance preferences. This was one of the ground-breaking ideas in finance. We will formally set up the investor’s portfolio choice problem and learn step-by-step how to solve for the optimal allocation and risky portfolio choice given a set of risky securities. You will also have an opportunity to apply these techniques to a numerical example. This module is slightly more technical than the others. Stick with it… you will not regret it!

Taught By

Arzu Ozoguz

Finance Faculty

Transcript

Hello, welcome back. The process of constructing an overall portfolio requires two steps. The first one is choosing a portfolio risky assets. In other words, selecting the composition of the risky portfolio given the risky assets you have available. Given the universe of risky assets. Step two, is deciding how much of your wealth you're going to invest in that risky portfolio and how much you're going to invest in the risk-free assets. So this is what we call the asset allocation decision. So, in this lecture we're going to use mean-variance optimization to answer these questions. In the process you're going to learn terms like means variance efficient portfolio, sharpe ratio, capital allocation line. And most importantly, you will understand the main intrusion and the insight behind optimal asset allocation decision. And then you can use this insight when you construct your own overall portfolio. And the underlying assumption that we are going to make is that you are risk overs like many of us and you have mean variance preferences. In other words, you want higher expected returns and you don't want a lot of risk. Now the amazing insight that will come out of this analysis is that regardless of what your attitude towards risk is. The best risky asset portfolio, the best risky portfolio, the answer to the first question, the first step. In fact, will be identical for all investors. In other words, there will be one optimal risky portfolio that is best for all investors, regardless of their preferences for risk. So really then, what will matter for each one of us is simply the answer to the second question. In other words, how much do you want to allocate to the risky portfolio versus risk free assets. So in this lecture, we're going to start with the second question and trust me, we will go back in, I'll make sure that I've convinced you about the first question, later on. But for now let's supposed that we have already identified what is this risky? What is actual risky portfolio is? And what all we need to do is decide, how much we want to missing this portfolio? And how much we want to miss in the risk-free assets? Okay, so remember we assumed that we already know the composition of this risky portfolio and what we're trying to do is decide and how we combine it with the risk-free assets, all right? So I'm going to look at the case of one risky asset and risk-free asset, so let's call the risky assets. So this could be a portfolio of stocks, it could be an index, right? So let's just assume that we know what it looks like and let me denote the return on this risky asset by rs, okay? And we also have a risk-free asset, maybe could be represented by treasury bills, right? A risk-free asset and let me denote the return on that as rf, for free, all right. And let's call W, the fraction invested in. The risky asset, right? So whatever that risky portfolio is, okay? So, now let's right down the expected return that we would get from combining this risky asset with the risk-free asset, right? So, well we know how to write that down, right? What is it going to be? Well, it's going to be the fraction we invested in the risky asset times the expected return on the risky asset plus 1 minus the weight times the return on the risk-free rate. All right, so, I can actually rewrite this, right? Using some algebra and you can write it as the risk-free rate plus the weight times the expected return on the risk asset minus the risk-free rate. That makes sense, right? What is our expected return on this portfolio? Well, it's whatever the risk-free rate is plus however much we invest in the risky asset times the excess return that we're getting on that risky asset or risky portfolio. What about the variance? Well, we know how to write that down, too, right? What is it going to be? Well, it's going to be the weight squared of the first asset, the risky asset, times the various of the risky asset, plus 1 minus the weight squared, right? The weight in the second asset times it's variance, 2 times the weights times the covariance between the three assets. But risk-free asset, of course, by definition is risk-free, right? So this is going to be zero, all right? And the covariance of a risk-free asset by with anything else is going to be also zero, all right. So those drop outs, basically, we're left with the variants of the portfolio being simply determined by how much we invest in the risky asset and its variants. Taking square root of both sides, what do you we have? Well, the volatility of this portfolio is going to be equal to the weight that we invest in the risky asset times its volatility, right? In other words, the risk of this portfolio will simply be determined by the volatility of the risky asset and how much we invest in the risky asset, right? So given the level of risk we desire, right, we chose, we can actually, so for the weight that will give us that as, and that's going to be this, okay? All right, so we got this two pieces, right? So let me now go ahead and plug this thing into the weight here, all right? What does that give me? Bear with me for a second, all right? It gives me the following expression, right, so, here is the expected return expression, all right? And, from the previous slides, got his expression for the weight, so let's now plug that in here. What do we get? Well. Times the expected excess return, all right? Well, let me re-write this a little more nicely, what you get is. Well, what you get is an expression right that relates to expected return on the portfolio to the volatility of the portfolio, right? In fact, this is what we call the Capital Allocation Line. Right? So, it's a line that we can draw in the mean variance, in the expected return volatility space, right? And it tells us the, all the risk and return combinations we can create from using this risky asset and a risk free assets, all right? And the intuition is, look at the intuition, what does it say? Well, it says that the expected return, is on this portfolio is going to be given by the risk free rate plus sum reward to risk ratio times risk, right? So let me draw that for you, right? So, I'm going to draw this capital allocation line, right? So remember it's lives in the expected return. Well, until it is space, right? So the intercept is the risk free rate, the Y X intersect is risk free, right? And it just tells us all the risk and return combinations that we can create, given the risky assets volatility and expected return. Right? This is the Capital Allocation Line. It gives us all this can return combinations that we can create from this two assets rise just like investment opportunities set. Now,the slope of this deliver right down the line equation again. All right, the slope of this line, we have a special name for it. This is what's called the sharp ratio, right? In other words, it tells you how much additional reward you get per unit of risk for holding in the risky asset. So it's the return premium that you get for investing in the risky asset per unit of risk. It's also sometimes called the reward to volatility ratio. Right? So this is the slope, which is called the Sharpe ratio, all right? Okay, so the question of course is, now that we have the investment opportunity sets that, this is of investing in a risk-free asset. And the risky asset and all the combinations, where along this line do you want to be? Well, how would you answer that question?

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